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Theorem bnj18eq1 29810
 Description: Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Assertion
Ref Expression
bnj18eq1

Proof of Theorem bnj18eq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj602 29798 . . . . . . . . . . 11
21eqeq2d 2481 . . . . . . . . . 10
323anbi2d 1370 . . . . . . . . 9
43rexbidv 2892 . . . . . . . 8
54abbidv 2589 . . . . . . 7
65eleq2d 2534 . . . . . 6
76anbi1d 719 . . . . 5
87rexbidv2 2888 . . . 4
98abbidv 2589 . . 3
10 df-iun 4271 . . 3
11 df-iun 4271 . . 3
129, 10, 113eqtr4g 2530 . 2
13 df-bnj18 29572 . 2
14 df-bnj18 29572 . 2
1512, 13, 143eqtr4g 2530 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   w3a 1007   wceq 1452   wcel 1904  cab 2457  wral 2756  wrex 2757   cdif 3387  c0 3722  csn 3959  ciun 4269   cdm 4839   csuc 5432   wfn 5584  cfv 5589  com 6711   c-bnj14 29565   c-bnj18 29571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4271  df-br 4396  df-bnj14 29566  df-bnj18 29572 This theorem is referenced by:  bnj1137  29876
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